The Pitfalls of Black Scholes:

The Pitfalls of Black Scholes:
How to Value Executive Stock Options Under AASB 2
The majority of ASX-listed companies utilise share-based payments
(SBPs) as a mechanism to retain and motivate key staff, and to align their
interests with shareholders. Executive stock options (ESOs),
performance shares and performance rights are common examples.
Consider an Example
As an executive is granted 100,000 ESOs with the following terms:
Share-Based Payments Have Exotic Features
1. The $1.20 exercise price on the options is marginally higher than
current share price of $1. That is, the options are granted out-ofthe-money.
The securities issued to executives are rarely “plain-vanilla”. Rather, they
are designed with exotic features to appropriately incentivise the recipient.
There are countless possibilities, but common examples include:
2. The options have a two-year vesting period. The recipient
forfeits the options if they leave the employment of the company
during the vesting period.

a time-based vesting period designed to retain key staff. The
recipient will not receive any securities granted to them if they
leave the employ of the company before completion of the vesting
period.
3. The options have a five-year total life. Therefore, after
completion of the two-year vesting period, there is a further threeyear window during which the holder may elect to exercise their
right to buy shares in the company.

market-based performance hurdles. For example, targets may be
specified in relation to share-price uplift or total shareholder return
(TSR) over a designated measurement period.
4. Half of the total options issued (50,000 options) are subject to an
EPS performance condition. Specifically, all 50,000 options vest
if EPS in either FY01 or FY02 reaches 30 cents per share,
otherwise none of these options vest.

non-market-based performance hurdles. For example, EPS growth
must be exceed a designated target over the measurement period.
The rationale for incorporating performance hurdles is obvious. In fact, in
many cases, it is desirable to design remuneration schemes using multiple
performance hurdles.
© Philip Gray 2011
5. The remaining half of the total options issued (a further 50,000
options) are subject to a TSR performance hurdle measured
over the two-year vesting period. Specifically, if TSR does not
reach 15%, none of the options vest. If TSR exceeds 25%, all of
the options vest. For TSR performance between 15% and 25%,
the number of options that vest increases linearly from zero to
50,000.
Accounting Requirements under AASB 2
It is relatively straight forward to design a remuneration scheme along
these lines. The challenge is to value the options.
AASB 2 requires that the fair value of the SBPs be measured as at grant
date and expensed over the life of the options. The fair value is the price
that would be expected in an arm’s length transaction between two
knowledgeable, willing parties (paragraph 17). Naturally, fair value takes
into account the terms and conditions upon which those SBPs were
granted (paragraph 16).
In the example given above, there are clearly some very important terms
and conditions that need to be factored into the valuation in order to
comply with AASB 2 Share Based Payments.
The Limitations of Black-Scholes
The Black-Scholes (BS) option pricing model was developed in 1973
shortly after plain-vanilla call options first began trading on the Chicago
Board Options Exchange. The BS formula is one of the most-famous
valuation models in modern finance. No doubt, every business-school
graduate would have at least heard of it. It is imperative, however, to
understand what the BS model can and can’t do.
The Black-Scholes formula determines the fair value of European-style
options written on stocks that do not pay dividends. Quite simply, that is all
it does. Recall that European-style options are those that cannot be
exercised before their expiry date. Given that the vast majority of options
that trade on exchanges around the world can be exercised at any point in
time (i.e., they are American-style options), the BS model does even
“work” for the most common type of option.
To further understand the limitations of the BS formula for valuing ESOs,
consider the following. The calculations that sit behind the BS formula
involve probabilistic assessments. The formula essentially looks at the
current share price ($1 in the above example), looks at the exercise price
($1.20), considers the volatility of the underlying stock, then estimates the
likelihood that share price will rise above the exercise price by the end of
the option’s life (five years).
How appropriate would it be to apply the BS formula to price the ESOs
detailed above? The answer, of course, is that the BS model is woefully
inadequate for this purpose!
For starters, the BS model simply assumes that the options are held
through to their five-year expiry then exercised if in-the-money and left to
lapse if out-of-the-money. BS cannot accomodate the fact that the ESOs
have a two-year vesting period, followed by a three-year window during
which the holder may elect to exercise early.
The more serious deficiencies of BS occur in the presence of performance
hurdles. With respect to the TSR hurdle, one needs to estimate what the
likely TSR will be over the two-year measurement period. This determines
how many of the 50,000 options will vest. The BS formula was never
designed to incorporate this level of complexity!
Similarly, the EPS hurdle requires an assessment of the likely EPS
outcomes in FY01 and FY02. If we recall that EPS (nor any other
accounting variable) is not even an input variable to the BS formula, it is
readily apparent that BS has no way to factor in the EPS performance
condition.
© Philip Gray 2011
In a nutshell, any attempt to apply the BS model to estimate the fair value
of the ESOs described above is misguided. Of course, we could ignore the
vesting period, ignore the fact that the ESOs might be exercised before
expiry, ignore the TSR performance hurdle and ignore the EPS
performance hurdle, and simply proceed to use the BS formula to obtain a
number. However, this number will reflect the value of a European call
option – which is not the security we need to value for AASB 2 reporting!
As such, any attempt to apply the BS model to value executive stock
options (especially when performance hurdles are involved) is unlikely to
be deemed an appropriate valuation approach under AASB 2. If used for
this purpose, we could be forgiven for thinking the acronym “BS” stands for
something else ...
How To Estimate the Fair Value of SBPs to Comply with AASB 2?
Since the BS model is of little use, how then can we estimate the fair value
of ESOs and other share-based payments? The most popular and
arguably only suitable approach is Monte Carlo simulation. While the
term “MC simulation” may be familiar, the procedure and logic for its
implementation are often less well understood.
Without going into technical details, the aim of MC simulation is to assess
the distribution of possible outcomes for a variable of interest. For the
above example, we require the distribution of our company’s TSR over the
two-year measurement period. If we conducted one million MC
simulations, we can then ascertain: (i) how many simulations produce a
TSR less than 15% in which case none of the options vest, (ii) how many
simulations produce a TSR exceeding 25% in which case all of the options
vest, and (iii) how many simulations produce a TSR at each point between
15% and 25%.
Similarly, a MC simulation approach can assess the distribution of possible
EPS outcomes in FY01 and FY02. Examining the frequencies from many
simulations allows quantification of the likelihood of the EPS performance
hurdle being satisfied.
The beauty of MC simulation is that it is extremely flexible. In fact, it is
sufficiently flexible to accomodate just about any performance criteria that
might be attached to SBPs.
As an example, many remuneration schemes incorporate a “relative”
performance based hurdle along the following lines. Options are granted
to an employee, but the number of options that ultimately vest is
determined by comparing our company’s TSR over the measurement
period to the TSR of a group of peer stocks. In some cases, the peers may
be direct competitors. Often the group of peer stocks comprises all
companies in the S&P/ASX200 index.
To incorporate this condition into the valuation of the options, we need to
estimate the distribution of possible TSR over the measurement period,
not only for our company, but also for each of the 200 stocks in the
S&P/ASX200 index. And since stocks tend to move together to a certain
extent, it is necessary to incorporate typical correlations between all
stocks. While this is a complicated task – and certainly an exercise well
beyond the scope of the BS model – the required distributions can be
assessed using MC simulation.
© Philip Gray 2011
Takeaways

AASB 2 requires estimation of the fair value of share-based
payments using an approach that takes into account their precise
terms and conditions.

Share-based payments are rarely so simple in their design that an
off-the-shelf formula like Black-Scholes will provide fair value.

Share-based payments are rarely so simple that off-the-shelf
software can be used to estimate fair value.

Increasingly, more flexible valuation approaches like Monte Carlo
simulation are being implemented to comply with AASB 2.
For more information on MC valuations for AASB 2 reporting, contact Dr
Philip Gray (0438 711 530).
Valuation Analytics was founded in January 2007 to serve a growing demand for
the independent valuation of the option component inherent in
executive/employee remuneration packages. We have completed over 60
valuations to assist ASX-listed companies to meet their annual reporting
requirements under AASB 2.
Our forte is Monte Carlo valuation, especially when performance hurdles are
based on TSR. Most of our business is repeat business. Most of our business is
referred to us from other industry participants.
© Philip Gray 2011
© Philip Gray 2011